PHYSICAL REVIEW D VOLUME 41, NUMBER 10

Cosmic strings in general relativity

A. K. Raychaudhuri Physics Department, Presidency College, Calcutta 700 073, India

(Received 6 October 1989)

This paper makes a survey of the study of gravitational fields of cosmic strings of different types. The Hiscock-Gott metric is given a new interpretation, the occurrence of a singularity in the case of global strings is explored, and the inadmissibility of the angle-deficit idea in some cases is pointed out.

I. INTRODUCTION

Spontaneous symmetry breaking in gauge theories may give rise to some topologically trapped regions of a false vacuum-domain walls, strings, or monopoles depending on the dimension involved. Some of these strings may have played a significant role in the structure formation in the Universe. Three types of strings will figure in our discussion: namely, the strings arising from a breaking of U(1) gauge symmetry and the global string arising from the breaking of a global U(1) symmetry as well as the su- perconducting string which has an axial-vector current.

There has been a fairly large amount of discussion on the gravitational field of different types of strings begin- ning with the work of vilenkinl who studied the linear- ized field equations of general relativity in the case of a gauge string. However an assumption of a weak field in the core of the string and its immediate neighborhood is not justified and further Vilenkin used an expression for the energy stress tensor of the string which was obtained in a somewhat ad hoc manner and is not correct in gen- eral.

Somewhat later isc cock^ and Gott3 claimed to give solutions of the exact general-relativity equations but again used the Vilenkin form of the energy stress tensor along with a further stipulation that the energy density is constant within the string. Using the Lagrangian of a U(1) gauge field and a complex scalar field we shall show that the Hiscock-Gott solution is not consistent with proper boundary conditions. We shall offer an alterna- tive interpretation to their metric as representing a homogeneous stationary magnetic universe.

Still later, interest shifted to the study of the exact field equations by computation and interesting results were ob- tained by ~ a r f i n k l e , ~ Laguna-Castillo and Matzner,' as well as Garfinkle and ~ a ~ u n a . ~ A somewhat novel inves- tigation was due to Frolov, Israel, and ~ n r u h . ' Without using any specific Lagrangian they explored the conse- quences of the symmetry assumptions (namely, static cy- lindrical symmetry together with a Lorentz boost along the symmetry axis) and the requirement that the fields pass over to that of a true vacuum at least asymptotically at radial infinity.

The global string involves only a complex scalar field. For such strings Harari and Skivies presented a solution of the linearized field equations neglecting the radial vari- ation of the scalar field outside the core of the string.

15 MAY 1990

(The field inside the core was not considered.) That the approach was not self-consistent became evident as the metric tensor components blew up at a finite radial dis- tance. Cohen and ~ a ~ l a n ~ considered the exact equa- tions of general relativity but still considered the scalar field to be independent of the radial coordinate outside the core of the string. In their solution, there was an unexplained singularity on the symmetry axis while the string appeared as a singular cylindrical surface of finite radius. That an undesirable singularity is unavoidable for the global string was later proved by Gregory1' who as- cribed it to a peculiarity of the energy stress tensor rather than to the slow falloff of the energy density with radial distance. She referred in this connection to the field of superconducting strings given by Moss and Poletti. ' I We shall show that Gregory's contention is not quite correct and some of the conclusions of Moss and Poletti are not justified.

It is the purpose of the present investigation to make a critical study of the previous works, bringing in the pro- cess a unified outlook for different strings and we highlight some points which have hitherto been over- looked.

11. THE FIELD EQUATIONS

The Lagrangian of a U( 1) gauge field along with a com- plex scalar field is

where

The potential term involves the constants h and 7 and the lowest-energy state occurs at the symmetry-broken value of 4=+7. With the assumptions of static cylindri- cal symmetry the metric is of the form

d s 2 = e A d t 2 - e B d z 2 - e C d 8 2 - d p 2 , (2)

where A , B, and C are functions of the radial coordinate p alone. We shall further assume A = B which means that the system admits a Lorentz boost along the symme- try axis Z . Cylindrical symmetry demands that 0 is an angle coordinate but one must confirm that by suitable analysis as we emphasize later.

We take

41 3041 @ 1990 The American Physical Society -

3042 A. K. RAYCHAUDHURI

where again X and P are functions of p alone. The above forms are suggested by the solutions of Nielsen and olesen'' for flat space. The nonvanishing components of the energy stress tensor T p v for a gauge string can now be written down using Eqs. (1) - (3) :

where we have numbered the coordinates t,p,z,8 as 0,1,2,3, respectively, with

and primes indicate differentiation with respect to r. For the global string, the vector field A p does not exist

and the expressions for Tpv are obtained from (4) - (6) by putting P= I . Again for a simple magnetic field, only the terms involving P' survive.

We have altogether four nontrivial equations coming from the equations of the gauge field and those of general relativity. They may be reduced to

The above are for the gauge string. For the global string equation (9 ) falls off and in the other equations one has to put P= 1 . For the magnetic field case equation (8) falls off, the right-hand side of ( 9 ) vanishes and only the terms involving P' survive in the right-hand side of (10) and ( 1 1 ) .

111. THE HISCOCK-GOTT METRIC

The Hiscock-Gott m e t r i ~ ~ > ~ was presented as a solution for an infinitely long gauge string. With proper choice of scale the metric is

and the energy stress tensor T p , is

The metric (12) was claimed to be matched with a true-vacuum field ( Tp,=O) at p=po with the metric

However, it is easy to see that (12)-(14) are incon- sistent with expressions (4) - (6) . From (131, (51, and (6 ) we obtain, with e Z A = 1 ,

In the true vacuum, Tp,=O gives, from (4 ) ,

If now X and P are continuous at the boundary, then from Eqs. ( 1 5 ) and (16) X' = P'=O just inside the string as well. However in that case from Eq. (41, T O , ( = T ~ , ) also vanishes just inside which makes a + co with the energy stress tensor vanishing throughout the string. If, on the other hand, one allows a discontinuity of the field vari- ables, then Tp , would blow up at the boundary. Thus the Hiscock-Gott metric cannot be accepted as a solution for the gauge string field.

However the Hiscock-Gott metric has interesting properties which have not been emphasized. First, how are we to interpret the energy stress tensor? The case T ' , = T ~ , =O has been investigated analytically by LinetI3 and numerically by Laguna-Castillo and ~ a t z n e r ~ who found that the corresponding metric is not that of Hiscock and Gott ( A'# 0 ) and the energy density is not constant. However a look at the energy stress tensor ex- pression shows that (13) follows if X =X1=O. P ' / K = f . The first two terms in the large parentheses on the right- hand sides in (4) - (6) then drop out and the third term mimics a cosmic constant and the fourth term may be in- terpreted as due to a Maxwellian magnetic field obeying sourcefree Maxwell's equation to which (9 ) reduces for X=O. There is no possibility of matching with a vacuum metric-apparently the solution covers all space. One may wonder what the "singularities" at K =O ( p = n .rra ) signify. However the spacetime given by (12) is complete- ly homogeneous and admits in all six Killing vectors. In addition to the translations along the t , z, and 0 axis and the Lorentz boost, it admits the additional Killing vec- tors

COSMIC STRINGS IN GENERAL RELATIVITY 3043

( a ) cl=cosO, c3= -a cotp/a sin0 , (17)

[The existence of six Killing vectors for the metric (12) was noted efrlier in a different context by Teixeira, Re- boucas, and Aman. 1 4 ]

Thus the appropriate interpretation of the Hiscock- Gott solution seems to be a homogeneous stationary universe with a magnetic field and cosmological term. In this, it may be compared with the magnetic universe of MelvinI5 which too is static and cylindrically symmetric with a magnetic field but is not spatially homogeneous and does not involve any cosmological term.

1V. SOME GENERAL CONSIDERATIONS

Both the global and the gauge strings of infinite length that we consider are characterized by cylindrical symme- try, static nature and a Lorentz boost along the symmetry axis. Further for any string, one expects that either abruptly at a finite radial distance the field is matched with a true vacuum or asymptotically at radial infinity TF,-0 as p- m . Thus in any case one should have a vacuum metric as p+ m . With the symmetries that we have mentioned there are only two distinct vacuum metrics: namely,

and

Of course, one may have simple scale changes along t, z, and 8 so that the metric components g,,, g,,, and gee are determined only up to arbitrary multipliers; the coordi- nate p has been normalized to measure proper lengths along the radial lines. That 8 is an angular coordinate in (19) may be justified by the consideration that the trans- formation

reduces (19) to the Minkowski form. However, for (20) there is no a priori reason for identifying 0 as an angular coordinate and the term cylindrical symmetry must be justified by a proper analysis in the particular case.

If there is a boundary to the string field at a finite r, then from (4) in the outside vacuum,

But X and P must be continuous at the boundary as oth- erwise X' and P' would blow up leading to an infinite en- ergy density. Also the continuity of normal stress T ' ~ further requires, from Eq. ( 5 ) , X1=P'=O inside the string as well. It then follows from (8) and (9) that X = l ,P=O everywhere and the field vanishes completely.

Thus passing over to the vacuum may take place only asymptotically as p- co. One may decide whether the metric will pass over to (19) or (20) by considering Eq. (10). As eC/2 goes to zero as p for p-0 (this is a condi- tion for 0 to be an angular coordinate and the origin to be a regular point) and for p- co, A' goes to zero faster

than 1 /p in the case of (19) and as 4/3p in the case of (20). Consequently K A ' vanishes for both p+O and p--+ co in the case of (19) but for (20), K A ' vanishes at p-0 and has a positive value for p- m . Hence from Eq. ( lo), we have

(x2-1) ' + 2eiAp" 0 [ - 2 I dr=O [for ( l 9 ) ] ,

a K

>O [for (20)] .

(21b)

For (21a) to be satisfied, one must have a gauge string in which both the vector field A , and the scalar field 4 ex- ists. Indeed the computational analysis of Laguna- Castillo and Matzner5 has shown that such solutions do exist. However apparently that seems to require a fine- tuning between the scalar and the vector fields.

In case only one field exists (21a) cannot apparently be satisfied, while if only the P field exists (21b) will be satisfied. This is the situation in Melvin's magnetic universe with the metricI5

If one makes the transformation

- 3 r 7=+-=p,

3

then for T >> 1,T = 3p and the line element (22) becomes

where we have made a scale change for t and z. That the global string satisfies neither (21a) nor (21b)

and the other limiting case of a magnetic field satisfies (21b) suggests that the gauge string fields also may pos- sess these peculiarities depending upon the initial condi- tion that one employs on the field. However that possi- bility has usually been overlooked and conclusions drawn from the metric (19) have been advanced as generally true for gauge strings. [Cf. the remarks in Frolov, Israel, and unruh7- he second alternative cannot represent a stringlike object. . . and we follow conventional practice in disregarding it."]

For iglobai string, outside the core, the radial depen- dence of the 4 field has usually been neglected, so that X = l . While this does not satisfy Eq. (8) as P= 1, one may wonder whether the metric will now pass over to the form (19) as (21a) is satisfied. However (21a) is only a necessary condition and not sufficient for passage to ( 19) -indeed Eq. ( 1 1 ) now becomes

which agrees neither with (19) nor (20). We shall take up the case of the global string in the next section.

A. K. RAYCHAUDHURI

V. THE GLOBAL STRING

For these strings, the symmetry breaking is global and one may take 4 to be of the form #=f (r)eie. It is presumed that outside the core of the string f ( r ) rapidly attains a constant value 1 7 7 Neglecting altogether the radial dependence o f f ( r ) Cohen and ~ a ~ l a n ~ gave the metric

as an exact solution of the Einstein equations. In the above u is the radial coordinate and y ,uo are constants. The metric has singularities both at u=O and u + m.

However the Ricci tensor tends to blow up as u + rn and remains finite for u -0. One thus identifies the singulari- - ty at u -+ rn as the string-nevertheless, the proper dis- tance from u=O to m is finite showing that the space is radially closed. The singularity at u=O can be under- stood if one makes the transformation

Then in the neighborhood of u=O (or p=O) the metric becomes (absorbing some constants by scale changes)

which is of the form (20) and associates a finite energy stress tensor with the axis p=O. The singularity is thus physical.

~ r e ~ o r ~ " has argued that the singularity must occur quite generally in the case of global strings. The follow- ing simple argument also indicates the inevitable oc- currence of a singularity. At sufficiently large values of p, where X = 1, Eq. (1 1 ) for a global string reduces to

where d is positive and f a constant. Thus K is bounded, whereas in the cylindrically symmetric field with the metric form (2) as p + rn one expects K to be unbounded if there is no singularity.

In a recent paper Gibbons, Ortiz, and ~ u i z ' ~ have confirmed the absence of regular solutions for static glo- bal strings with cylindrical symmetry. However they consider a singular solution to "correspond to a physical global string."

VI. THE SUPERCONDUCTING STRING

For the superconducting string, there is a current along the symmetry axis and consequently a magnetic field in the 8 direction. The geometry is no longer boost invariant; however, we expect that at radial infinity where the energy stress tensor tends to vanish the metric

will be transformable to the general Kasner form

where the constants a ,P , y are subject to the conditions

Equations (26) indicate that a,/?, y cannot be all positive and each lies between - 1 and + 1.

In their investigations on the gravitational field of the superconducting string, Moss and ~ o l e t t i " assumed that at large distances from the core of the string, the energy stress tensor is dominated by the magnetic field. Thus for large distances, they effectively obtained the metric for a simple magnetic field which was (apart from some arbi- trary constants) the same as that obtained by Ray- chaudhuri" in his study of static electromagnetic fields in general relatively.

The metric is

In fact ~ u k h e r j e e ' ~ was the first to obtain a metric which he considered to be due to an infinite straight wire carrying current. Mukherjee's metric is a transformation of the metric (27) with p = 2 . However the interpretation was questioned by onn nor'^ on the ground that at large distances the magnetic flux was not the same as that of a classical line current.

To appreciate Bonnor's criticism, let us calculate the energy stress tensor on the basis of metric (27). We get

So that, for r >> 1 and p > 0,

writing

p=r 'p2+p+l)

where p is the "proper distance" from the axis for large values of r, we obtain, for the magnetic flux B,

B - 1

P 1+~p/p2+p+l) .

Thus B does not fall off a s p - ' which is the classical re- sult. However for a static metric which is regular at p-, rn , the integral

which is identified as the gravitational mass per unit coordinate length of the axis, must converge. Hence for the electromagnetic field T O , must vanish faster than p-2.

(Note that for Kasner-type metrics v m -p.) Hence a

4 1 - COSMIC STRINGS IN GENERAL RELATIVITY 3045

magnetic field which has the classical behavior ( B ap-' ) would lead to an energy diverging logarithmically and thus bring in a singularity.

The case of the global string may be mentioned in this connection. There one has the diagonal components of T ' , - ( ~ ~ ~ ) - ' for large p (cf. Cohen and ~ a ~ l a n ~ ) , so that the gravitational mass assumes the form - S ( g e e ) - ' g M d p and with the metric (251, the in- tegrand is . Hence with lrl 5 1, the integral diverges. Thus contrary to the conjecture of Gregory," the falling off of the energy density at the rate p-2y does play a part in the occurrence of a singularity in the global string field.

The existence of the three Killing vectors a/&, a/az, and a / a 6 gives rise to three integral relations. Thus if cP is a hypersurface orthogonal Killing vector, we get by, straightforward calculation,

where

Eq. (29) being a consequence of the hypersurface ortho- gonality of r. For large values of r such that only the highest power term of r needs to be retained one can transform (27) to the form (25):

ds2=p2adt2 -dp2-p2~dz2-c2p2~d~2 , (30)

where

The constant c is normalized by the condition that, on the axis, gee- -p2d02, which ensures also the angular nature of the coordinate 6.

With the coordinate system of Eq. (30), l p = 6 : (a=0,2,3). Then $ = x ", D =gas. Hence,

The above expressions (differ somewhat from those given in Ref. 11) indicate that ,u plays the crucial role in the integrated values of the energy tensor, rather than the constants C 1 , C2 appearing in (27).

VII. CONCLUDING REMARKS

It has been customary practice to discuss the angle deficit in the case of string fields. When the asymptotic field is of the form

as in the case of gauge ~ t r i n ~ s , ~ - ~ the transformation

reduces (31) to the Minkowski form and one case says that @'=KO plays the role of the angle coordinate in cy- lindrical polar coordinates in a Euclidean field. But the domain of 6 ' = 2 n K f 277. Hence comes the idea of a deficit angle = 2 ~ ( 1-K). The occurrence of this deficit angle was investigated by ~ r e ~ o r ~ . ~ ' Taking the metric in the form

[note Eq. (4) in Gregory's paper] she showed that a - r and r +a constant value for r + m if the energy stress tensor components fall off sufficiently rapidly as r-+ m . She then concluded the asymptotically conical nature of spacetime. However we have already shown that in many cases the asymptotic metric tends to the form (20) (e.g., the Melvin universe) or (30) (in the case of supercon- ducting strings). Thus in these cases Gregory's argument ceases to apply and gee is not a constant multiplied by p2. Indeed the deficit angle has s o m e t i m ~ b e e n defined as 2 ~ - ( a L /ap),,, where L E Si"Z/ lgoe(d6. Thus for general metrics of the form (20) or (30) this definition gives the angle deficit as just 2n. This result seems to in- dicate that in these cases one cannot talk of an angle deficit-indeed it becomes auestionable whether one can

R ; = - - [ ( g a a ) , l ~ ~ / g a a ~ , l ( a=0 ,2 ,3 . ) . at all consider 6 to be an angle coordinate at p+ cc . 2V'T;;/ In the general fields of Kasner type, there is either a

, u ,

Using the field equations, collapse in the Z or 0 direction but g, the determinant of the metric tensor, remains nonsingular and spacetime is asymptotically flat as indicated b; the vanishing of the Riemann-Christoffel tensor.

ACKNOWLEDGMENTS

This work was supported by the Indian National Sci- ence Academy (INSA).

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